158 XIII INTEGRATION z e K and u(z) <£ ^ u(z). (By the Hahn-Banach theorem, the points (z, £) belonging to S are those with the following property: if h is any continuous affine linear function on E x R such that /z(y, u(y)) ^ 0 for all y e K, then /z(z, £) ;> 0. (Use Problem 5.) (b) Show that if u is strictly convex in K, then u(±) < u(z) for all nonextremal points z in K. (c) Show that, for each point z e K, there exists a positive measure ^ on K having z as barycenter and such that u(z) == I u(y) <s^(y). (Apply Problem 2(b) to the closed convex hull S of the compact set G in E x R.) Deduce from Problem 4 that u(z) = v(z) almost everywhere in K, with respect to ft. 8. Let E be a separable real Frechet space and K a compact convex subset of E. (a) Show that the set M of extremal points of K is the intersection of a sequence of open sets of K, and therefore universally measurable. (In the space K x K x [0,1], consider the set of triples (z', z", A) such that z' ^ z" and 0 < A < 1, and its image K — M under the mapping (z', z"', A) i—>Az' + (1 — A)z".) (b) Prove that every point of K is the barycenter of a positive measure ft on K, with total mass 1, such that ft(K — M) = 0 (" Choquef s theorem ")• (Use Problems 6 and 7.) 9. Extend the results of Problems 2 to 8 to the case of a convex and vaguely compact subset K of M+(X), where X is a compact space (cf. Section 12.15, Problem 13). 11. THE SPACES L1 AND L2 Throughout the remainder of this chapter we shall make the following convention: the function (x, y)\-+xy will be extended to R x R by putting xy = 0 whenever one of the factors is 0 and the other is ± oo (of course, this extension to R of the multiplication on R is not continuous). With this convention, we have x(yz) = (xy)z for all x, y, z<= R, and the relations x ^ y and z ^ 0 imply xz <j yz. If /is any mapping of X into K, we put /** / /** \ 1/2 (13.11.1) N1(/) = \f\dft, N2(/) (1 3.11 .2) For all mappings/ ^ 0 and g^O ofX into R, we have (13.11 .2.1) Np(/+ g) ^ Np(/) + Np(#) (p = 1, 2). This has already been proved for p = 1 (13.5.6). For p = 2 we shall first prove (with the above convention for products in M) the generalized Cauchy- Schwarz inequality (13.11.2.2) Nx(/<7) £ N2(/)N2(#). This inequality is true if one of the factors on the right-hand side is zero. If for example N2(#) =0, then g2 is negligible and therefore so is g (13.6.3)on K' with total mass 1